Variational principles for BS dimension under amenable group actions

TL;DR

This paper explores the properties of BS dimension and BS packing dimension in the context of amenable group actions, establishing fundamental relations and principles including Bowen's equation, variational principles, and analogues of Billingsley's theorem.

Contribution

It introduces new variational and inverse variational principles for BS dimensions under amenable group actions, extending the theoretical framework of dynamical systems.

Findings

Derived Bowen's equation relating BS packing dimension to topological pressure

Established variational and inverse variational principles for BS dimensions

Proved an analogue of Billingsley's theorem for BS packing dimension

Abstract

In this manuscript, we focus on the investigation of the BS dimension and BS packing dimension under amenable group actions. Firstly, we obtain a Bowen's equation which illustrate the relation of BS packing dimension to the packing topological pressure under amenable group actions. Moreover, we establish the variational principle and inverse variational principle for BS dimension and BS packing dimension under amenable group actions. Finally, we also get an analogue of Billingsley's type theorem for BS packing dimension under amenable group actions.